Equiform bundle motions in \(E_3\) with spherical trajectories. I
In 1906 and 1908 Bricard and Borel published two long papers in which they determined Euclidean motions with spherical trajectories [\textit{R. Bricard}, J. l'École Polytechnique, II. Sér. 11, 1-93 (1906; JFM 37.0705.04); \textit{E. Borel}, Mém. Sav. Étrangers, Paris 33, 1-128 (1908; JFM 39.0749.02)]. Based on these papers the authors determine equiform bundle motions which have spherical trajectories. They introduce a 10-dimensional parameter space \(P(W)\) in which a 6-dimensional surface \(B_6\) represents the condition that a point remains on a sphere during an equiform motion. Furthermore, an embedding of the motion group into a 10-dimensional projective space \(P(V)\) is used to give an interpretation of the sphere condition as an orthogonality relation between the points of \(P(V)\) and \(P(W)\). It is shown that the images of sphere conditions given by any motion are on a linear intersection of \(B_6\). Because of this fact, linear subspaces of \(P(W)\) and their intersection with \(B_6\) are studied. The case that a line is contained in \(B_6\) results in either one point having a circle as trajectory or the points of a circle or a line of the moving system are constrained to move on spheres or in planes. The case of the intersection of a 2-dimensional linear space with \(B_6\) yields a one parameter motion which moves the points of a bundle plane on spherical trajectories. [For Part II, see the following review Zbl 0907.53009].
scientific article; zbMATH DE number 1208007